| L(s) = 1 | + 2.45·2-s − 0.192·3-s + 4.03·4-s + 5-s − 0.473·6-s − 4.67·7-s + 5.00·8-s − 2.96·9-s + 2.45·10-s + 2.77·11-s − 0.777·12-s + 0.198·13-s − 11.4·14-s − 0.192·15-s + 4.22·16-s − 17-s − 7.27·18-s − 4.33·19-s + 4.03·20-s + 0.899·21-s + 6.82·22-s + 3.53·23-s − 0.963·24-s + 25-s + 0.487·26-s + 1.14·27-s − 18.8·28-s + ⋯ |
| L(s) = 1 | + 1.73·2-s − 0.111·3-s + 2.01·4-s + 0.447·5-s − 0.193·6-s − 1.76·7-s + 1.76·8-s − 0.987·9-s + 0.776·10-s + 0.837·11-s − 0.224·12-s + 0.0550·13-s − 3.06·14-s − 0.0497·15-s + 1.05·16-s − 0.242·17-s − 1.71·18-s − 0.994·19-s + 0.902·20-s + 0.196·21-s + 1.45·22-s + 0.738·23-s − 0.196·24-s + 0.200·25-s + 0.0956·26-s + 0.221·27-s − 3.56·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 3 | \( 1 + 0.192T + 3T^{2} \) |
| 7 | \( 1 + 4.67T + 7T^{2} \) |
| 11 | \( 1 - 2.77T + 11T^{2} \) |
| 13 | \( 1 - 0.198T + 13T^{2} \) |
| 19 | \( 1 + 4.33T + 19T^{2} \) |
| 23 | \( 1 - 3.53T + 23T^{2} \) |
| 29 | \( 1 - 4.11T + 29T^{2} \) |
| 31 | \( 1 + 8.04T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + 2.31T + 41T^{2} \) |
| 43 | \( 1 - 2.02T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 2.41T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 2.79T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 73 | \( 1 + 3.73T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 + 2.07T + 89T^{2} \) |
| 97 | \( 1 + 9.51T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.12378551895662226458848230098, −6.61424637837558383687592067693, −6.25281224998104628837215292456, −5.60271855682880892305849139395, −4.89575073261655232434958741614, −3.92830632485030420900980352358, −3.30733022042958002439986983987, −2.80480889222063591629262045016, −1.80018164872123315350296754307, 0,
1.80018164872123315350296754307, 2.80480889222063591629262045016, 3.30733022042958002439986983987, 3.92830632485030420900980352358, 4.89575073261655232434958741614, 5.60271855682880892305849139395, 6.25281224998104628837215292456, 6.61424637837558383687592067693, 7.12378551895662226458848230098
